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Tag Archives: modules
Commutative Algebra 58
We have already seen two forms of unique factorization. In a UFD, every non-zero element is a unique product of irreducible (also prime) elements. In a Dedekind domain, every non-zero ideal is a unique product of maximal ideals. Here, we … Continue reading
Posted in Advanced Algebra
Tagged annihilators, associated primes, localization, module division, modules, supports
4 Comments
Commutative Algebra 35
Noetherian Modules Through this article, A is a fixed ring. For the first two sections, all modules are over A. Recall that a submodule of a finitely generated module is not finitely generated in general. This will not happen if we constrain … Continue reading
Posted in Advanced Algebra
Tagged flat modules, hilbert basis theorem, ideals, modules, noetherian, projective modules
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Commutative Algebra 28
Tensor Products In this article (and the next few), we will discuss tensor products of modules over a ring. Here is a motivating example of tensor products. Example If and are real vector spaces, then is the vector space with … Continue reading
Posted in Advanced Algebra
Tagged bilinear maps, distributive property, modules, tensor product, universal properties
2 Comments
Commutative Algebra 21
Exact Sequences When studying homomorphisms of modules over a fixed ring, we often encounter sequences like this: where each is a homomorphism of modules. This sequence may terminate (on either end) or it may continue indefinitely. Note on indices: usually … Continue reading
Posted in Advanced Algebra
Tagged additive functors, exact sequences, functors, modules, short exact sequences
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Commutative Algebra 10
Algebras Over a Ring Let A be any ring; we would like to look at A-modules with a compatible ring structure. Definition. An –algebra is an -module , together with a multiplication operator such that becomes a commutative ring (with 1); multiplication … Continue reading
Posted in Advanced Algebra
Tagged algebras, generated submodules, homomorphism, modules, quotient modules, rings, submodules
5 Comments
Commutative Algebra 9
Direct Sums and Direct Products Recall that for a ring A, a sequence of A-modules gives the A-module where the operations are defined component-wise. In this article, we will generalize the construction to an infinite collection of modules. Throughout this article, let denote … Continue reading
Posted in Advanced Algebra
Tagged direct products, direct sums, modules, rings, universal properties
5 Comments
Commutative Algebra 8
Generated Submodule Since the intersection of an arbitrary family of submodules of M is a submodule, we have the concept of a submodule generated by a subset. Definition. Given any subset , let denote the set of all submodules of M containing … Continue reading
Posted in Advanced Algebra
Tagged free modules, generated submodules, homomorphism, modules, quotient modules, rings, submodules
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Commutative Algebra 7
Modules Having dipped our toes into algebraic geometry, we are back in commutative algebra. Next we would like to introduce “linear algebra” over a ring A. Most of the proofs should pose no difficulty to the reader so we will … Continue reading
Posted in Advanced Algebra
Tagged ideals, linear algebra, module homomorphism, modules, rings, submodules
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Idempotents and Decomposition
Let R be a general ring, not necessarily commutative. An element x∈R is said to be idempotent if x2 = x. Note An endomorphism f of an R-module M (i.e. ) is an idempotent if and only if f is a projection, i.e. M = ker(f) ⊕ im(f) and f … Continue reading
Posted in Notes
Tagged blocks, idempotents, indecomposable modules, modules, primitive idempotents
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Tensor Product over Noncommutative Rings
Following the earlier article on tensor products of vector spaces, we will now look at tensor products of modules over a ring R, not necessarily commutative. It turns out we have to distinguish between left and right modules now. Indeed recall … Continue reading
Posted in Notes
Tagged bimodules, hom functor, left-exact, modules, right-exact, tensor products
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Mathematics and Such 